\chapter{Final remarks: our contribution and some open questions}
\label{chap:conclusion}

May the principle be used to represent the landscape evolution
process in time and space using \aclp{DEM}?
\bigskip

Would a multi-objective framework be suitable to take 
advantage of the existing knowledge and to assess the trade-offs 
among different simplified version of the \enquote{optimality} 
criterion in order to find the proper expression for 3D modeling?
\bigskip

The previous two questions were written in the introduction as the
goal of this thesis. Now, after having commented the results, are
we able to answer?

It is important to remind that, for reaching the goals, we
detailed a multi-criteria optimization framework which describes
river and landscape evolution over a 3D spatial domain.
 
After the simulations run, the results obtained and the analysis
performed, we are ready to assess the value of the framework
proposed. The effort in the analysis and reduction of the aspects
around modeling landscape evolution under multiple principle
allows to assess the different criteria. We found that some of
them are conflicting, validating the hypothesis of opposite
phenomena captured by the \ac{LAP} as in \cite{rodriguez:1992}.
Studying this conflict means understanding which drivers of
landscape evolution are captured by each formulation.
Several tradeoffs between criteria formulations should be relate
to hydrogeological features such as different degrees of
equilibrium between soil erodibility and channel incision or to
geological constraint such as bedrock properties. This should give
more awareness of the meaning of the \ac{LAP}.

In addition to the identification of the conflict, creating the
framework \ie a complete methodology, tools for studying the
phenomenon and assessment strategies for evaluating the
outputs forced us to deepen and expand the questions on landscape
evolution modeling. We had to provide mathematical formulations
for the simplified versions of the \ac{LAP}), in a suitable form
for \acp{DEM}.

Therefore, by considering each part of the framework separately:
\begin{itemize}
  \item as for its core, \ie the model, some important
  improvements and tests were put in practice. In particular, with
  respect to the main modeling reference of this work, \ie
  \citeauthor{paik:2011}'s \ac{GLE} model \cite{paik:2011}, a
  spatial interpolator was integrated in the model, showing
  promising results with respect to the simulations of slopes and
  the study of river profiles.
  Moreover, the rainfall parameter of the model was tested;
  \item as for the optimization phase, it was executed rigorously,
  adopting top performing state of the art algorithms. 
  Additionally, the performances of these algorithm and the
  advantage of using them in spite of simple random search, were
  quantified and assessed;
  \item the outputs analysis was based on many hydrological
  indexes, consistently with other important works. Moreover, the
  dimensional burden of the results was tackled by implementing
  clustering and statistical, allowing them to be globally
  considered.
\end{itemize}

Finally, it is possible to say, as an answer, that the framework
we developed is suitable to study the criteria and evaluating
their trade-offs, and is a complete tool for developing further
studies.

\section{Our contribution to the field of study}
This thesis contributes to the field studying landscape and river
evolution as follows:
\begin{itemize}
  \item as for modeling purposes, it is the first attempt that
  provides a model to reproduce the 3D features of landscapes and
  river networks based on multiple criteria optimization;
  \item as for sistem understanding purposes, it deepened the
  knowledge about the different formulations of least action
  principle and the way they affect the features of river networks
  developing on landscapes.
\end{itemize} 
The deliverables of this work are, therefore, good basis for
future researches towards the right formulation of the \ac{LAP}
driving landscape and river evolution.
For this reason, many improvements and open questions are
suggested, in the next, last section.

\section{Proposed improvements and open questions}
Given the largest amount of topics touched by this thesis, some
relevant issues are proposed to future development.

The first idea is to better setup and configurate the \ac{IDW}
interpolator. The results of \myExpThree are promising but they
also show significant limitations. This is something we are
already working on with the perspective of validiting the
framework with the comparison with a real case study. In case
additional problems should arise, would the implementation of a
different interpolator be better?

At the same time the optimization part of the framework can be
improved by performing sensitivity analysis and trials on
different algorithms parameterizations. The goal is to find the
best parameterization and the best performing algorithms to
reduce the amount of time and resources needed for the
optimization. Alternatively, algorithms which do it automatically,
like Borg or Amalgam can be used. Also a parallelization of model
and algorithm could take advantage of \ac{HPC} and lighten
the computational costs.

A part from this improvements, the open issues become more
complicated. The first fundamental one is related to time. In
order to really explore the concept of evolution, the time
dimension should be integrated in the model. In this case,
hydrological parameters, like rainfall, would assume values
directly comparable to the ones measured in natural events.
Moreover, the \ac{DEM} configuration would be relatable to time
itself and the rate of change in its elevation would be linked to
natural erosion and deposition rates.

Questions related to this issue that need to be answered can be:
\begin{itemize}
  \item what is the resilience of the system over time? is it
  stable somehow? what kind of equilibrium does it reach?
  \item what is the right scale, both in term of time and space,
  for reproducing the action caused by phenomena explained by the
  optimality principle?
  \item further exploiting the concepts of control theory, is a
  control policies a proper structure for aggregating the  
  criteria over time?
\end{itemize}

The second issue, is related to conflict. Given the fact that
conflicting behaviour was shown by the different formulations of
the \ac{LAP}, how is it possible to enforce the theoretical
interpretation of such a conflict? Moreover, is the conflict
really related to different features in the landscapes? Or does it
express a form of multifinality in landscape evolution? How the
time dimension affect the conflict shown? And more simply, what
are the variations in values of the objectives to be considered as
significant with respect to natural systems? All these questions
are promising topics for research. They can also be a theorical
base to study the principle of equifinality in landscape
evolution, \eg to analyze different \ac{DEM} shapes that lead to
the same optimized landscape after the operations described in the
framework.

Among the biggest issues, one last topic is worth to be mentioned:
our model is based on \acp{DEM}, given the advantages explained in
the text. The landscape is discretized on a cell basis, it is made 
evolve and then river networks are extracted. 
An alternative approach would be to base the analysis on a finer
discretizations which would better describe river networks, like
\acp{TIN}. These structures can also allow to directly optimize
river networks without losing the $z$ dimension, in spite of
landscapes. Would such an approach better describe the 3D features
of river networks?

\bigskip
		
Other improvements and open questions are given below in sparse
order.

On the modeling side:
\begin{itemize}
  \item Planchon's \ac{DF} algorithm should be improved, in order
  to better recognize real depressions from false depressions,
  caused by abnormal isolated small peaks and, consequently,
  better respect the mass constraint. Are there algorithms which
  can do it?
  \item the sensitivity of the model to different hydrological
  parameters, \ac{DEM} dimension and shape should be evaluated;
  \item the choice of the threshold for river network extraction
  must be better defined, according to state of the art methods.
\end{itemize}
		
On the results analysis tools:
\begin{itemize}
  \item a stronger study on the correlation between naturality
  indexes and \ac{DEM} features should be performed, in order to
  understand if they properly describe the 3D features. Moreover,
  the definition of their natural ranges should be enforced.
  Are there any more suitable and correct indexes?
  \item the comparison between synthetic landscapes and natural
  ones should be enforced and supported by data.
  Therefore, which set up of the model allows it to output
  landscapes really comparable to natural ones? What is the best
  method for proving that?
\end{itemize}
	
Challenges appear to be many because our framework is a completely
new perspective on a complex issue.
\medskip


\begin{flushright}{

\slshape    
Plus grand est l'obstacle, plus grande est la gloire de le surmonter
\footnote{The greater the obstacle, the more glory in overcoming it.}} \\ 
Molière wrote some time ago.
\end{flushright} 
